All about poi and noi

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This page sets out to explain all you need to now about the relative clause markers poi and noi.

It does not contain separate explanations for pe, ne, po'u and no'u, because those are nothing more than shortcuts of poi and noi combined with specific predicates. Everything that applies to poi and noi directly applies to those cmavo, too. No separate explanations are required. For a conversion table see the bottom of this page.

The examples on this page will consist of a Lojban phrase or expression, a rendering in logic notation, and an English translation.

poi and noi in general

poi and noi introduce relative clauses, restrictive and non-restrictive respectively. They help us narrow down referent sets or allow us to provide additional information about a given sumti. For both poi and noi, their syntactic position and environment make a big difference as to what exactly they achieve in a given sentence. We will look at each and every possible case below.

A general note about inner quantifiers

It should be noted that so-called inner quantifiers (the ones between gadri and selbri) aren't true logical quantifiers in that they don't count true bridi, but merely enumerate the number of referents of a sumti. The presence or absence of an inner quantifier has no effect on the behaviour of relative clauses, be it outer or inner relative clauses. What a relative clause means is solely dependent on whether it is a restrictive (poi) or non-restrictive (noi) one, and whether it is an inner (lo broda NOI) or outer (lo broda ku NOI) relative clause.

PA da poi

[...]

What does a quantifier do? A quantifier counts how many times or for how many things a bridi is true, or, put another way, a quantifier counts true bridi.

[...]

domain

anything -> anybody

examples

A useful thing to know is that a sentence of the form su'o da poi broda cu brode is truth-functionally equivalent to su'o da ge broda gi brode:

su'o da ge cribe gi blabi [math][\exists x]\colon \text{cribe}(x) \land \text{blabi}(x)[/math]"There exists some X such that X is both a bear and white."

For ro da poi broda cu brode the following equivalence holds instead:

ro da poi ke'a broda cu brode

equals

ro da zo'u ga nai da broda gi da brode

For example:

Lojban[math]logic[/math]"English."

which means the same as:

Lojban[math]logic[/math]"English."

PA <selbri>

By definition, PA da poi ke'a broda can be abbreviated PA broda. This is a handy shortcut that saves space, variables, and terminators. Apart from the different grammar, PA broda behaves identical to PA da poi ke'a broda in all contexts.

PA <sumti>

poi also appears "invisibly" any time a sumti carries an outer quantifier.

A quantifier [math]\square[/math] counts for how many things that are among the referents of <sumti> the bridi is true, or in other words, it tells you that if you were to substitute each member for <sumti>, then the bridi would be true exactly [math]\square[/math] times.

This is the formula that is evoked any time an outer quantifier occurs in a bridi:

Whence the universal quantifier? In simple terms, we know that noi is non-restrictive, which means that its presence must not have any impact on the number of true bridi that the quantifier [math]\square[/math] counts. The only logical way in which a relative clause could make a statement without restricting (i.e. selecting a sub-set) is if the predicate of the noi-clause applies to every member of a given referent-set.

Prefixing a na ku to a bridi is always a good way to gain insights about the scope interactions within the bridi. How does such a negation differ between a poi sentence and a noi sentence?

In the noi version, we know that nothing broda, but all things still brode, because noi makes a separate claim.

In the poi version, we know have a much weaker claim, namely that nothing satisfies both brode and broda together ([math]\text{broda}(x) \land \text{brode}(x)[/math]), while nothing is said about how many things satisfy broda, and how many satisfy brode.

<sumti> NOI

<sumti> noi

Let's begin this section with the simplest case first, <sumti> noi. This is noi in its most basic habitat. noi makes an incidental claim about the head noun which is separate from the main bridi. The general syntax of <sumti> noi is this:

While it's possible to translate mi noi prami do cu tolstace into English as "I love you and am being dishonest", this is not recommended, because it could suggest that noi can be translated into gi'e, which it cannot, as becomes apparent when the na ku is added back in:

na ku mi prami do gi'e tolstace[math]\neg ( \text{prami}(\text{mi}, \text{do}) \land \text{tolstace}(\text{mi}))[/math]"It's not the case that I both love you and am being dishonest." (i.e., at least one of those two is false)

Therefore it's best to always consider the noi clause as entirely independent from the main claim, lest one could make wrong assumptions.

<sumti> poi

The previous section that <sumti> noi is very straightforward. It has an obvious meaning and its logic is easy to grasp. <sumti> poi on the other hand might be less obvious. In the section about da poi we saw poi restricting the domain of a quantifier. But here we have no quantifier, so what is the poi, the restrictive relative clause marker, to restrict in <sumti> poi?

What does this mean? It means that <sumti> poi ke'a brode creates a new sumti whose referents are those that are both among <sumti> and satisfy brode. Since there is no quantifier whose domain could be restricted, the poi has to perform a restriction directly on the sumti. It does this by selecting referents from among the sumti which satisfy a certain predicate, namely the predicate inside the relative clause.

It is important to note that precisely all referents that are among <sumti> and satisfy brode are among the referents of the resulting expression, not just some of the referents. This is called maximality.

Looking at the logic notation, we can see that the constant [math]c[/math] is only stated to be among [math]c1[/math] and to brode, but not that all referents among [math]c1[/math] that brode are among [math]c0[/math], even though that's what the poi version implies. The following always holds with outer poi:

However, since this takes up so much space, we will omit the part in the middle with the understanding that maximality is implied in all examples involving outer poi. This is to keep things easier to read.

Let's look at some examples:

Lojban[math]logic[/math]"English."

[...]

Lojban[math]logic[/math]"English."

[...]

<sumti> poi with apparently singular <sumti>

Certain kinds of sumti tend to be interpreted as singular, like mi (presupposing a single speaker here), or perhaps a sumti formed with zo. What effect does using poi on a sumti like this have, and what does this tell us about poi?

[...]

Selbri-based descriptions

Relative clause placements

A selbri-based description is a sumti made from a gadri plus a selbri, e.g. lo plise. With selbri-based descriptions there are three different places a relative clause (abbreviated <rel>) can go:

lo <rel> broda <rel> ku <rel>

By definition, a relative clause that appears between gadri and selbri is equivalent to one that appears after the ku:

lo <rel> broda

is equivalent to:

lo broda ku <rel>

Because they are equivalent we will only analyse the case where the relative clause follows ku to avoid duplication.

If the relative clause is placed between selbri and ku then the meaning changes compared to the other two cases above.

Inner relative clauses - lo <selbri> NOI

lo <selbri> poi

[...]

Quite a different case is lo broda poi brode, where the relative clause is placed between selbri and ku. Here, poi semantically acts even before lo to define the initial predicate of which lo then extracts the x1. The resulting predicate has the same referent set as the conjunction of the initial predicate and the relative clause.

lo broda poi brode

A poi in this position is exactly equivalent to je poi'i:

lo broda je poi'i ke'a brode

And this is equivalent to:

lo poi'i ke'a broda gi'e brode

Since the inner poi merely defines the initial referents of the sumti, it has no bearing on outer quantifiers. lo broda poi brode ku is really just a specific case of lo <selbri> ku.

For example:

lo plise poi xunre

Yields a sumti that refers to red apples. Let's examine the logic in a full sentence:

What happens, when poi is outside the ku? Then we get a case we've already covered: <sumti> poi broda (we recall that <sumti> poi broda is the same as lo me <sumti> je broda, which is the same as lo poi'i ke'a me <sumti> gi'e broda). But we will consider this case in more detail later.

(not even a negation could cause a difference here, because the sentence contains nothing but constants)

As we've seen above, an inner poi clause attaches directly to the selbri, in a sense restricting the referent set of the predicate by replacing the single predicate with a conjunction of it and another predicate.

In other words, the poi in lo broda poi brode creates a new predicate [math]H[/math] that has the same referent set as [math]F \land G[/math].

In this sense, poi performs a restrictive roll, even in the absence of any logical quantifiers. noi, however, is the non-restrictive counterpart. It, too, will try to attach directly to the predicate, but without restricting the referent set of the resulting predicate. This means that it does not create a new predicate, or, perhaps more accurately, the referent set of [math]F \land G[/math] in the case of noi is equivalent to that of the lone predicate [math]F[/math].

The nuance here is more subtle than with quantifiers and applies directly to the universe of discourse. Let's build an analogue between the quantified noi/poi distinction we already understand, and the inner noi/poi distinction to see the parallels:

As covered much earlier, the noi implies a universal quantifier, because that's the only logical way that a conjunction can have the same referent set as the predicates in isolation. A similar thing could be said about inner noi, let's compare:

lo broda poi brode cu brodi (replace with actual words!!)

lo poi'i ke'a broda gi'e brode cu brodi"English"

But in keeping with the theme of creating a new predicate, let's introduce a me'au:

This could be written simpler if Lojban had a non-restrictive version of je (something I've proposed it should have), but it doesn't.

Thus, we can conclude that the previous example implies a universe of discourse where [math]F[/math] and [math]F \land G[/math] have the same referent set, or put another way, in which for all things, being broda entails being brode.

No such implication occurs in the poi version.

Again: Inner poi creates a new predicate that has the same referent set as the conjunction of the two conjoined predicates.

Inner noi does not create a conjunction, but rather comments on the predicate saying that being broda also means being brode (in the current universe of discourse).

Outer relative clauses - lo <selbri> ku NOI

lo <selbri> ku poi

If you have read everything above, then this section contains no truly new information.

lo <selbri> ku is a case of <sumti>, and we already know how <sumti> interacts with quantifiers and relative clauses. There isn't anything new to understand here, but we'll look at an example regardless:

With this example, the two interpretations give the same result, the same number of delicious apples. However, the next example shows the inadequacy of interpretation 1:

lo pulji cu arrest ro lo panono panja'o ku poi sruri lo dinju

(Note by the way, that the same sentence with inner poi would translate to English as "The police arrested each of the 100 building-surrounding demonstrators", i.e., all 100 both demonstrated and surrounded the building)

(Note that the noi version differs significantly from a hypothetical poi version. With noi we know that exactly three apples are taken. With poi we know that three red apples are taken, but not whether any non-red apples are taken)

Again, both interpretations yield the same number of red apples taken by the speaker (three red apples), but they get to that result by different mechanisms. Which one is correct? Interpretation 1 parallels Interpretation 1 of the poi, and interpretation 2 parallels interpretation 2. We saw that interpretation two of poi was better, does the same apply to noi?

Let's consider noi in the context of the demonstrators that surrounded the building:

Note that this does not mean that it's necessarily the case that nobody else among the demonstrators surrounded the building. It does however say that each single arrested demonstrator surrounded the building individually, which is not the intended meaning, and likely a very bizarre situation. Maybe that's why the police felt the need to arrest them.

This says that in fact all 100 demonstrators were building-surrounders. Thus, any 20 that the police could arrest automatically are building-surrounders also. Additionally, interpretation 2 again does what interpretation 1 does not: account for plural quantification.

But what if we do want to say that what interpretation 1 said (minus the distributivity)?. We might not want to claim each demonstrater a building-surrounder. We might want to say that each arrested demonstrator was a building-surrounder, no, that each arrested demonstrator was among building-surrounders. Well, we can:

lo mi <selbri>

la and relative clauses

ma NOI

First, let's recap how the question word ma (as well as all the other question words) behave.
ma always has top-scope, no matter how deeply nested it appears in a sentence. (1) and (2) are equivalent:

(1) do djica lo nu mi zukte ma
"You want me to do what?"
"What do you want me to do?"
(2) ma poi'i do djica lo nu mi zukte ke'a
"What is such that you want me to do it?"
"What is it you want me to do?"

However, if we wish to restrict the referent pool of the question word, the familiar method of using poi (as in (3)) may not be semantically accurate -- the desired expansion of (3) is (4), but the actual meaning is (5). The difference might be subtle and difficult to spot, but it's there, and it means that we should find a new way to say "which/what".

Both methods leave something to be desired. Even if noi has the right semantics, it still requires a terminator and a ke'a, while using mo can be vague (since it's a tanru) and it can be awkward to use.

Therefore, it might be a good idea to create a new gadri X (in selma'o LO) that has the semantics of (4) but keeps the wh in-situ capability of (3). In other words, ko'a broda X brode would be defined as ma brode gi'e poi'i ko'a broda ke'a.

It remains to be decided how that new gadri should be spelled.

Summary

[...]

There are only two rules to learn, two rules, which explain every possible combination of quantifiers and relative clauses.